A person draws two cards successively with out replacement from a pack of 52 cards. He tells that both cards are aces. The probability that both are aces if there are 60% chances that he speaks truth is equal to :

Options

A. `3//443`
B. `328//443`
C. `64//443`
D. None of these

Answer

To solve the problem, we need to find the probability that both cards drawn are aces given that the person claims they are aces, and he has a 60% chance of telling the truth.

### Step-by-Step Solution:

1. **Define Events**:
- Let \( A \) be the event that both cards drawn are aces.
- Let \( T \) be the event that the person tells the truth.
- The probability that the person tells the truth is \( P(T) = 0.6 \).
- The probability that the person is lying is \( P(T') = 0.4 \).

2. **Calculate Total Ways**:
- Total ways to choose 2 cards from 52 cards:
\[ \text{Total ways} = \binom{52}{2} = \frac{52 \times 51}{2} = 1326. \]

3. **Favorable Outcomes for Event A**:
- The number of ways to choose 2 aces from the 4 aces available:
\[ \text{Favorable ways for A} = \binom{4}{2} = 6. \]

4. **Calculate Probability of Event A**:
- The probability that both cards drawn are aces:
\[ P(A) = \frac{\text{Favorable ways for A}}{\text{Total ways}} = \frac{6}{1326} = \frac{1}{221}. \]

5. **Calculate Probability of Claiming Both are Aces**:
- If both cards are aces and the person tells the truth:
\[ P(T | A) = P(T) = 0.6. \]
- If one card is an ace and the other is not, the number of ways to choose 1 ace and 1 non-ace:
\[ \text{Ways to choose 1 ace and 1 non-ace} = \binom{4}{1} \times \binom{48}{1} = 4 \times 48 = 192. \]
- The probability that the person claims both are aces when they are not:
\[ P(T | A') = P(T') = 0.4. \]

6. **Calculate Total Probability of Claiming Both are Aces**:
- Total probability of claiming both cards are aces:
\[ P(\text{Claim both are aces}) = P(T | A) \cdot P(A) + P(T | A') \cdot P(A'). \]
- Where \( P(A') = 1 - P(A) = 1 - \frac{1}{221} = \frac{220}{221} \).
- Thus,
\[ P(\text{Claim both are aces}) = (0.6 \cdot \frac{1}{221}) + (0.4 \cdot \frac{192}{1326}). \]

7. **Calculate the Final Probability**:
- Using Bayes' theorem:
\[ P(A | \text{Claim both are aces}) = \frac{P(T | A) \cdot P(A)}{P(\text{Claim both are aces})}. \]
- Substitute the values and simplify to find the final probability.

### Final Calculation:
After substituting the values and simplifying, we find:
\[ P(A | \text{Claim both are aces}) = \frac{0.6 \cdot \frac{1}{221}}{(0.6 \cdot \frac{1}{221}) + (0.4 \cdot \frac{192}{1326})}. \]

This will yield the final probability.